I'd like to solve this equation for $A = B$ where $B = I$, which represents 3 systems of 3 linear equations, for $a, b, c, d, e, f$, without writing LinearSolve 3 times. What is a simple way to accomplish this?

@b.gatessucks If A == I then also A^-1 == I. Thus he just wants to find a,b,c,d,e,frather than A^-1.
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ArtesSep 9 '12 at 0:44

@T.Webster Could you explain what you'd expected from an answer, if existing ones were not satisfactory ?
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ArtesSep 15 '12 at 23:48

@Artes Sorry I have been away and thanks for your answer.
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T. WebsterSep 17 '12 at 8:08

The question related to finding the left inverse of some matrix $A$. But a solution I found from my linear algebra course is to use Gaussian elim. (RowReduce) with the augmented matrix $[A I]$.
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T. WebsterSep 17 '12 at 8:13

2 Answers
2

As the documentation says : LinearSolve[m,b] finds an x which solves the matrix equation m.x == b, i.e. in your case it finds x such that A.x == B (Dot[A,x] == B). However your task is to find A solving an adequate system of 9 linear equations for 6 variables a,b,c,d,e,f knowing that B is an IdentityMatrix. You are trying to solve an overdetermined system of linear equations and there could exist any solutions only if certain compatibility conditions were satisfied.

For your task use simply Solve :

Solve[A == B, {a, b, c, d, e, f}]

{}

or

Reduce[A == B, {a, b, c, d, e, f}]

False

This means that there are no solutions, i.e. the above equation is contradictory.
You could use Variables[A] instead of specifying variables {a, b, c, d, e, f}.

Consider a different matrix equation where we have 4 unknowns and 4 independent equations e.g. :

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